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Theorem recexprlemm 7380
Description:  B is inhabited. Lemma for recexpr 7394. (Contributed by Jim Kingdon, 27-Dec-2019.)
Hypothesis
Ref Expression
recexpr.1  |-  B  = 
<. { x  |  E. y ( x  <Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A ) ) } ,  {
x  |  E. y
( y  <Q  x  /\  ( *Q `  y
)  e.  ( 1st `  A ) ) }
>.
Assertion
Ref Expression
recexprlemm  |-  ( A  e.  P.  ->  ( E. q  e.  Q.  q  e.  ( 1st `  B )  /\  E. r  e.  Q.  r  e.  ( 2nd `  B
) ) )
Distinct variable groups:    r, q, x, y, A    B, q,
r, x, y

Proof of Theorem recexprlemm
StepHypRef Expression
1 prop 7231 . . 3  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
2 prmu 7234 . . 3  |-  ( <.
( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  ->  E. x  e.  Q.  x  e.  ( 2nd `  A ) )
3 recclnq 7148 . . . . . . 7  |-  ( x  e.  Q.  ->  ( *Q `  x )  e. 
Q. )
4 nsmallnqq 7168 . . . . . . 7  |-  ( ( *Q `  x )  e.  Q.  ->  E. q  e.  Q.  q  <Q  ( *Q `  x ) )
53, 4syl 14 . . . . . 6  |-  ( x  e.  Q.  ->  E. q  e.  Q.  q  <Q  ( *Q `  x ) )
65adantr 272 . . . . 5  |-  ( ( x  e.  Q.  /\  x  e.  ( 2nd `  A ) )  ->  E. q  e.  Q.  q  <Q  ( *Q `  x ) )
7 recrecnq 7150 . . . . . . . . . . . 12  |-  ( x  e.  Q.  ->  ( *Q `  ( *Q `  x ) )  =  x )
87eleq1d 2183 . . . . . . . . . . 11  |-  ( x  e.  Q.  ->  (
( *Q `  ( *Q `  x ) )  e.  ( 2nd `  A
)  <->  x  e.  ( 2nd `  A ) ) )
98anbi2d 457 . . . . . . . . . 10  |-  ( x  e.  Q.  ->  (
( q  <Q  ( *Q `  x )  /\  ( *Q `  ( *Q
`  x ) )  e.  ( 2nd `  A
) )  <->  ( q  <Q  ( *Q `  x
)  /\  x  e.  ( 2nd `  A ) ) ) )
10 breq2 3899 . . . . . . . . . . . . 13  |-  ( y  =  ( *Q `  x )  ->  (
q  <Q  y  <->  q  <Q  ( *Q `  x ) ) )
11 fveq2 5375 . . . . . . . . . . . . . 14  |-  ( y  =  ( *Q `  x )  ->  ( *Q `  y )  =  ( *Q `  ( *Q `  x ) ) )
1211eleq1d 2183 . . . . . . . . . . . . 13  |-  ( y  =  ( *Q `  x )  ->  (
( *Q `  y
)  e.  ( 2nd `  A )  <->  ( *Q `  ( *Q `  x
) )  e.  ( 2nd `  A ) ) )
1310, 12anbi12d 462 . . . . . . . . . . . 12  |-  ( y  =  ( *Q `  x )  ->  (
( q  <Q  y  /\  ( *Q `  y
)  e.  ( 2nd `  A ) )  <->  ( q  <Q  ( *Q `  x
)  /\  ( *Q `  ( *Q `  x
) )  e.  ( 2nd `  A ) ) ) )
1413spcegv 2745 . . . . . . . . . . 11  |-  ( ( *Q `  x )  e.  Q.  ->  (
( q  <Q  ( *Q `  x )  /\  ( *Q `  ( *Q
`  x ) )  e.  ( 2nd `  A
) )  ->  E. y
( q  <Q  y  /\  ( *Q `  y
)  e.  ( 2nd `  A ) ) ) )
153, 14syl 14 . . . . . . . . . 10  |-  ( x  e.  Q.  ->  (
( q  <Q  ( *Q `  x )  /\  ( *Q `  ( *Q
`  x ) )  e.  ( 2nd `  A
) )  ->  E. y
( q  <Q  y  /\  ( *Q `  y
)  e.  ( 2nd `  A ) ) ) )
169, 15sylbird 169 . . . . . . . . 9  |-  ( x  e.  Q.  ->  (
( q  <Q  ( *Q `  x )  /\  x  e.  ( 2nd `  A ) )  ->  E. y ( q  <Q 
y  /\  ( *Q `  y )  e.  ( 2nd `  A ) ) ) )
17 recexpr.1 . . . . . . . . . 10  |-  B  = 
<. { x  |  E. y ( x  <Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A ) ) } ,  {
x  |  E. y
( y  <Q  x  /\  ( *Q `  y
)  e.  ( 1st `  A ) ) }
>.
1817recexprlemell 7378 . . . . . . . . 9  |-  ( q  e.  ( 1st `  B
)  <->  E. y ( q 
<Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A
) ) )
1916, 18syl6ibr 161 . . . . . . . 8  |-  ( x  e.  Q.  ->  (
( q  <Q  ( *Q `  x )  /\  x  e.  ( 2nd `  A ) )  -> 
q  e.  ( 1st `  B ) ) )
2019expcomd 1400 . . . . . . 7  |-  ( x  e.  Q.  ->  (
x  e.  ( 2nd `  A )  ->  (
q  <Q  ( *Q `  x )  ->  q  e.  ( 1st `  B
) ) ) )
2120imp 123 . . . . . 6  |-  ( ( x  e.  Q.  /\  x  e.  ( 2nd `  A ) )  -> 
( q  <Q  ( *Q `  x )  -> 
q  e.  ( 1st `  B ) ) )
2221reximdv 2507 . . . . 5  |-  ( ( x  e.  Q.  /\  x  e.  ( 2nd `  A ) )  -> 
( E. q  e. 
Q.  q  <Q  ( *Q `  x )  ->  E. q  e.  Q.  q  e.  ( 1st `  B ) ) )
236, 22mpd 13 . . . 4  |-  ( ( x  e.  Q.  /\  x  e.  ( 2nd `  A ) )  ->  E. q  e.  Q.  q  e.  ( 1st `  B ) )
2423rexlimiva 2518 . . 3  |-  ( E. x  e.  Q.  x  e.  ( 2nd `  A
)  ->  E. q  e.  Q.  q  e.  ( 1st `  B ) )
251, 2, 243syl 17 . 2  |-  ( A  e.  P.  ->  E. q  e.  Q.  q  e.  ( 1st `  B ) )
26 prml 7233 . . 3  |-  ( <.
( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  ->  E. x  e.  Q.  x  e.  ( 1st `  A ) )
27 1nq 7122 . . . . . . . 8  |-  1Q  e.  Q.
28 addclnq 7131 . . . . . . . 8  |-  ( ( ( *Q `  x
)  e.  Q.  /\  1Q  e.  Q. )  -> 
( ( *Q `  x )  +Q  1Q )  e.  Q. )
293, 27, 28sylancl 407 . . . . . . 7  |-  ( x  e.  Q.  ->  (
( *Q `  x
)  +Q  1Q )  e.  Q. )
30 ltaddnq 7163 . . . . . . . 8  |-  ( ( ( *Q `  x
)  e.  Q.  /\  1Q  e.  Q. )  -> 
( *Q `  x
)  <Q  ( ( *Q
`  x )  +Q  1Q ) )
313, 27, 30sylancl 407 . . . . . . 7  |-  ( x  e.  Q.  ->  ( *Q `  x )  <Q 
( ( *Q `  x )  +Q  1Q ) )
32 breq2 3899 . . . . . . . 8  |-  ( r  =  ( ( *Q
`  x )  +Q  1Q )  ->  (
( *Q `  x
)  <Q  r  <->  ( *Q `  x )  <Q  (
( *Q `  x
)  +Q  1Q ) ) )
3332rspcev 2760 . . . . . . 7  |-  ( ( ( ( *Q `  x )  +Q  1Q )  e.  Q.  /\  ( *Q `  x )  <Q 
( ( *Q `  x )  +Q  1Q ) )  ->  E. r  e.  Q.  ( *Q `  x )  <Q  r
)
3429, 31, 33syl2anc 406 . . . . . 6  |-  ( x  e.  Q.  ->  E. r  e.  Q.  ( *Q `  x )  <Q  r
)
3534adantr 272 . . . . 5  |-  ( ( x  e.  Q.  /\  x  e.  ( 1st `  A ) )  ->  E. r  e.  Q.  ( *Q `  x ) 
<Q  r )
367eleq1d 2183 . . . . . . . . . . 11  |-  ( x  e.  Q.  ->  (
( *Q `  ( *Q `  x ) )  e.  ( 1st `  A
)  <->  x  e.  ( 1st `  A ) ) )
3736anbi2d 457 . . . . . . . . . 10  |-  ( x  e.  Q.  ->  (
( ( *Q `  x )  <Q  r  /\  ( *Q `  ( *Q `  x ) )  e.  ( 1st `  A
) )  <->  ( ( *Q `  x )  <Q 
r  /\  x  e.  ( 1st `  A ) ) ) )
38 breq1 3898 . . . . . . . . . . . . 13  |-  ( y  =  ( *Q `  x )  ->  (
y  <Q  r  <->  ( *Q `  x )  <Q  r
) )
3911eleq1d 2183 . . . . . . . . . . . . 13  |-  ( y  =  ( *Q `  x )  ->  (
( *Q `  y
)  e.  ( 1st `  A )  <->  ( *Q `  ( *Q `  x
) )  e.  ( 1st `  A ) ) )
4038, 39anbi12d 462 . . . . . . . . . . . 12  |-  ( y  =  ( *Q `  x )  ->  (
( y  <Q  r  /\  ( *Q `  y
)  e.  ( 1st `  A ) )  <->  ( ( *Q `  x )  <Q 
r  /\  ( *Q `  ( *Q `  x
) )  e.  ( 1st `  A ) ) ) )
4140spcegv 2745 . . . . . . . . . . 11  |-  ( ( *Q `  x )  e.  Q.  ->  (
( ( *Q `  x )  <Q  r  /\  ( *Q `  ( *Q `  x ) )  e.  ( 1st `  A
) )  ->  E. y
( y  <Q  r  /\  ( *Q `  y
)  e.  ( 1st `  A ) ) ) )
423, 41syl 14 . . . . . . . . . 10  |-  ( x  e.  Q.  ->  (
( ( *Q `  x )  <Q  r  /\  ( *Q `  ( *Q `  x ) )  e.  ( 1st `  A
) )  ->  E. y
( y  <Q  r  /\  ( *Q `  y
)  e.  ( 1st `  A ) ) ) )
4337, 42sylbird 169 . . . . . . . . 9  |-  ( x  e.  Q.  ->  (
( ( *Q `  x )  <Q  r  /\  x  e.  ( 1st `  A ) )  ->  E. y ( y 
<Q  r  /\  ( *Q `  y )  e.  ( 1st `  A
) ) ) )
4417recexprlemelu 7379 . . . . . . . . 9  |-  ( r  e.  ( 2nd `  B
)  <->  E. y ( y 
<Q  r  /\  ( *Q `  y )  e.  ( 1st `  A
) ) )
4543, 44syl6ibr 161 . . . . . . . 8  |-  ( x  e.  Q.  ->  (
( ( *Q `  x )  <Q  r  /\  x  e.  ( 1st `  A ) )  ->  r  e.  ( 2nd `  B ) ) )
4645expcomd 1400 . . . . . . 7  |-  ( x  e.  Q.  ->  (
x  e.  ( 1st `  A )  ->  (
( *Q `  x
)  <Q  r  ->  r  e.  ( 2nd `  B
) ) ) )
4746imp 123 . . . . . 6  |-  ( ( x  e.  Q.  /\  x  e.  ( 1st `  A ) )  -> 
( ( *Q `  x )  <Q  r  ->  r  e.  ( 2nd `  B ) ) )
4847reximdv 2507 . . . . 5  |-  ( ( x  e.  Q.  /\  x  e.  ( 1st `  A ) )  -> 
( E. r  e. 
Q.  ( *Q `  x )  <Q  r  ->  E. r  e.  Q.  r  e.  ( 2nd `  B ) ) )
4935, 48mpd 13 . . . 4  |-  ( ( x  e.  Q.  /\  x  e.  ( 1st `  A ) )  ->  E. r  e.  Q.  r  e.  ( 2nd `  B ) )
5049rexlimiva 2518 . . 3  |-  ( E. x  e.  Q.  x  e.  ( 1st `  A
)  ->  E. r  e.  Q.  r  e.  ( 2nd `  B ) )
511, 26, 503syl 17 . 2  |-  ( A  e.  P.  ->  E. r  e.  Q.  r  e.  ( 2nd `  B ) )
5225, 51jca 302 1  |-  ( A  e.  P.  ->  ( E. q  e.  Q.  q  e.  ( 1st `  B )  /\  E. r  e.  Q.  r  e.  ( 2nd `  B
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1314   E.wex 1451    e. wcel 1463   {cab 2101   E.wrex 2391   <.cop 3496   class class class wbr 3895   ` cfv 5081  (class class class)co 5728   1stc1st 5990   2ndc2nd 5991   Q.cnq 7036   1Qc1q 7037    +Q cplq 7038   *Qcrq 7040    <Q cltq 7041   P.cnp 7047
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4003  ax-sep 4006  ax-nul 4014  ax-pow 4058  ax-pr 4091  ax-un 4315  ax-setind 4412  ax-iinf 4462
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ne 2283  df-ral 2395  df-rex 2396  df-reu 2397  df-rab 2399  df-v 2659  df-sbc 2879  df-csb 2972  df-dif 3039  df-un 3041  df-in 3043  df-ss 3050  df-nul 3330  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-int 3738  df-iun 3781  df-br 3896  df-opab 3950  df-mpt 3951  df-tr 3987  df-eprel 4171  df-id 4175  df-iord 4248  df-on 4250  df-suc 4253  df-iom 4465  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-rn 4510  df-res 4511  df-ima 4512  df-iota 5046  df-fun 5083  df-fn 5084  df-f 5085  df-f1 5086  df-fo 5087  df-f1o 5088  df-fv 5089  df-ov 5731  df-oprab 5732  df-mpo 5733  df-1st 5992  df-2nd 5993  df-recs 6156  df-irdg 6221  df-1o 6267  df-oadd 6271  df-omul 6272  df-er 6383  df-ec 6385  df-qs 6389  df-ni 7060  df-pli 7061  df-mi 7062  df-lti 7063  df-plpq 7100  df-mpq 7101  df-enq 7103  df-nqqs 7104  df-plqqs 7105  df-mqqs 7106  df-1nqqs 7107  df-rq 7108  df-ltnqqs 7109  df-inp 7222
This theorem is referenced by:  recexprlempr  7388
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